2009/07/25 by Dinesh Deshpande, Deshpande, Dinesh · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P10 #msc:55P48 #msc:55S12
paper · pdf · doi:10.48550/arxiv.0907.4412
9 pages
arxiv created 2009/07/25 · arxiv updated 2009/12/01
Let Ratk be the space of based holomorphic maps from S2 to itself of degree k. Let betak denote the Artin's braid group on k strings and let Bbetak be the classifying space of betak. Let Ck denote the space of configurations of length less than or equal to k of distinct points in R2 with labels in S1. The three spaces Ratk, Bbeta2k, Ck are all stably homotopy equivalent to each other. For an odd prime p, the Fp-cohomology ring of the three spaces are isomorphic to each other. The F2-cohomology ring of Bbeta2k is isomorphic to that of Ck. We show that for all values of k except 1 and 3, the F2-cohomology ring of Ratk is not isomorphic to that of Bbeta2k or Ck. This in particular implies that the HF2-localization of Ratk is not homotopy equivalent to HF2-localization of Bbeta2k or Ck. We also show that for k >= 1, Bbeta2k and Bbeta2k+1 have homotopy equivalent HF2-localizations.